### Location of fixed points in the presence of two cycles

#### Abstract

Any orientation-preserving homeomorphism of the plane having a two

cycle has also a fixed point. This well known result does not

provide any hint on how to locate the fixed point, in principle it

can be anywhere. J. Campos and R. Ortega in {\it Location of fixed

points and periodic solutions in the plane} consider the class of

Lipschitz-continuous maps and locate a fixed point in the region

determined by the ellipse with foci at the two cycle and

eccentricity the inverse of the Lipschitz constant. It will be

shown that this region is not optimal and a sub-domain can be

removed from the interior. A curious fact is that the ellipse

mentioned above is relevant for the optimal location of fixed

point in a neighbourhood of the minor axis but it is of no

relevance around the major axis.

cycle has also a fixed point. This well known result does not

provide any hint on how to locate the fixed point, in principle it

can be anywhere. J. Campos and R. Ortega in {\it Location of fixed

points and periodic solutions in the plane} consider the class of

Lipschitz-continuous maps and locate a fixed point in the region

determined by the ellipse with foci at the two cycle and

eccentricity the inverse of the Lipschitz constant. It will be

shown that this region is not optimal and a sub-domain can be

removed from the interior. A curious fact is that the ellipse

mentioned above is relevant for the optimal location of fixed

point in a neighbourhood of the minor axis but it is of no

relevance around the major axis.

#### Keywords

Ellipse; Lipschitz-continuous homeomorphism; two cycle; fixed point

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